You think structure isn't the issue but it probably is. Just doubling everything up is no guarantee that you and your family will be safe. Please google FEMA 320 and there you can download the publication and safe room plans for Residents and Small Businesses. Your safe room size with wood framing is an option in the plans. Pay attention to the connectors required between all the framing members to transfer the load from the joists through the walls and into the foundation. You cannot just nail together the framing like typical construction or it will fail in the event of an EF5 tornado. Also 2 layers of 3/4" plywood is not enough to stop a 15 lb 2x4 at 100 MPH, which is the missile design criteria for safe rooms. You also need to add a layer of steel and use a specific nailing pattern. This information is included in the plans reference above. Also ventilation is important, don't let the other post make you think otherwise. Recommended ventilation and detailing how to do it is also in the plans reference above. Also foundation requirements are in the publication. Now the door, you will want to purchase a door that is designed to meet the criteria in FEMA 361, the is the guideline for Community Saferooms. It is the same design criteria as FEMA 320, but doors are labeled as 361 and not 320. They are expensive, but prices have been dropping, and you will probably have to order as I have not seen any retailers that sell them. Do not make the mistake of buying the heaviest steel door at Lowe's, it will fail during an event. If you have questions about FEMA 320, FEMA has a hotline you can call specifically about building a safe room in your house. Please note the information above is for a safe room that is designed for an EF5 tornado, if you modify the criteria I would recommend having a structural engineer with experience in safe rooms reviewing your changes. Otherwise your safety may be jeopardized during an EF5. Sincerely, The Structural Engineer that designed the safe rooms for FEMA 320.
P.S. Notify your local Emergency Management office that you have a safe room and where it is located so they can search for you, a properly designed door could open either way and have a whistle in there to signal for help. The FEMA publications have a list of recommended items to keep in the safe room.